 
Summary: MULTILEVEL SOLVERS FOR UNSTRUCTURED SURFACE MESHES
BURAK AKSOYLU , ANDREI KHODAKOVSKY , AND PETER SCHR ĻODER
Abstract. Parameterization of unstructured surface meshes is of fundamental importance in many applications
of Digital Geometry Processing. Such parameterization approaches give rise to large and exceedingly illconditioned
systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strate
gies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsen
ing to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies
using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hi
erarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid
preconditioner.
Key words. multilevel preconditioning, multigrid, hierarchical basis multigrid, BPX, computer graphics, un
structured surface mesh, surface parameterization, harmonic weights, mean value weights, mesh coarsening.
AMS subject classifications. 65Y20, 65F10, 65N55, 65D18, 65M50
1. Introduction. Unstructured triangle meshes which approximate surfaces of arbitrary
topology (genus, number of boundaries, number of connected components) appear in many
application areas. Examples range from isosurfaces extracted [54] from volumetric imaging
sources and scientific simulations [62] to surfaces produced through range scanning tech
niques (e.g., [10, 51]) in areas as varied as historical preservation, reverse engineering, and
entertainment. These meshes can be quite detailed: 100, 000 samples, i.e., point positions
on the surfaces, are quite common with many datasets ranging into the millions and some
